arXiv subjects

Daniel Drimbe

Publications and source records attributed to Daniel Drimbe.

21 records · Page 2Linked to original sources

arXiv2018

W$^*$-superrigidity for coinduced actions

We prove W$^*$-superrigidity for a large class of coinduced actions. We prove that if $Σ$ is an amenable almost-malnormal subgroup of an infinite conjugagy class (icc) property (T) countable group $Γ$, the coinduced action $Γ\curvearrowright X$ from an arbitrary probability measure preserving action $Σ\curvearrowright X_0$ is W$^*$-superrigid. We also prove a similar statement if $Γ$ is an icc non-amenable group which is measure equivalent to a product of two infinite groups. In particular, we obtain that any Bernoulli action of such a group $Γ$ is W$^*$-superrigid.

math.OA↗

arXiv2017

Prime II$_1$ factors arising from irreducible lattices in products of rank one simple Lie groups

We prove that if $Γ$ is an icc irreducible lattice in a product of connected non-compact rank one simple Lie groups with finite center, then the II$_1$ factor $L(Γ)$ is prime. In particular, we deduce that the II$_1$ factors associated to the arithmetic groups $\text{PSL}_2(\mathbb Z[\sqrt{d}])$ and $\text{PSL}_2(\mathbb Z[S^{-1}])$ are prime, for any square-free integer $d\geq 2$ with $d\not\equiv 1 mod{4}$ and any finite non-empty set of primes $S$. This provides the first examples of prime II$_1$ factors arising from lattices in higher rank semisimple Lie groups. More generally, we describe all tensor product decompositions of $L(Γ)$ for icc countable groups $Γ$ that are measure equivalent to a product of non-elementary hyperbolic groups. In particular, we show that $L(Γ)$ is prime, unless $Γ$ is a product of infinite groups, in which case we prove a unique prime factorization result for $L(Γ)$.

math.OA↗

arXiv2015

Cocycle superrigidity for coinduced actions

We prove a cocycle superrigidity theorem for a large class of coinduced actions. In particular, if $Λ$ is a subgroup of a countable group $Γ$, we consider a probability measure preserving action $Λ\curvearrowright X_0$ and let $Γ\curvearrowright X$ be the coinduced action. Assume either that $Γ$ has property (T) or that $Λ$ is amenable and $Γ$ is a product of non-amenable groups. Using Popa's deformation/rigidity theory we prove $Γ\curvearrowright X$ is $\mathcal U_{fin}$-cocycle superrigid, that is any cocycle for this action to a $\mathcal U_{fin}$ (e.g. countable) group $\mathcal V$ is cohomologous to a homomorphism from $Γ$ to $\mathcal V.$

math.OA↗